The arctan calculator returns the angle whose tangent is the number you enter. It differs from its two siblings in a way that shapes everything about how it is used: arctangent is the only inverse trigonometric function defined for every real number. There is no domain restriction to check, no rejected input, no "outside −1 to 1" error. You can hand it 0.001 or 10,000,000 and it will answer.
What it gives up in exchange is reach. Its range is (−π⁄2, π⁄2) — open at both ends, meaning −90° and 90° are approached but never attained. Arb Digital keeps this page separate from the arcsin calculator and arccos calculator because that unbounded-domain, open-range combination creates a completely different set of practical problems, chief among them the quadrant question that the two-argument form on this page exists to solve.
What This Arctan Calculator Does
In single-value mode, enter a tangent — usually a slope, a gradient, or a ratio of two legs of a right triangle — and get the angle in degrees and radians, with the exact form at the standard values. In two-argument mode, enter the y and x components separately and the calculator uses atan2, the function that reads both signs and returns the true direction anywhere in the full 360°.
Both modes also report the angle expressed on a 0° to 360° scale, which is what bearings, headings and polar coordinates usually want. The difference between the two modes is not cosmetic. Enter y = −3 and x = −4 and atan2 returns −143.13°, pointing down and to the left. Compute the ratio yourself as −3 ÷ −4 = 0.75 and feed that to plain arctan, and you get 36.87° — up and to the right, the exact opposite direction.
How to Use It
- Choose the mode. Use atan2 whenever you have two components and the direction matters. Use single-value mode when you only have a ratio and the angle is known to be acute.
- Enter your value or your y and x components. Signs are significant in atan2 mode and are what make it work.
- Pick degrees or radians. Degrees is the default; every programming language's atan2 returns radians.
- Read the principal value, which always lies strictly between −90° and 90° in single-value mode.
- Use the 0°–360° figure when you need a bearing or a polar angle rather than a signed value.
The Formula: How It's Calculated
Tangent is sin θ ÷ cos θ. As θ climbs towards 90°, cosine shrinks towards zero, and the ratio grows without limit — tan 89° is 57.29, tan 89.9° is 572.96, tan 89.99° is 5,729.6. It never stops growing and never arrives at a value for 90°. Read backwards, that is exactly why arctangent accepts any real number: every real number really is the tangent of some angle short of 90°.
The branch chosen for the inverse is the one from −90° to 90°, exclusive at both ends. y = arctan(x) means tan y = x with y strictly inside (−π⁄2, π⁄2). Working the default: arctan(1) asks which angle has equal rise and run, and the answer is 45°, or π⁄4. Push the input up and the output creeps towards the asymptote — arctan(1000) = 89.9427°, arctan(1,000,000) = 89.99994° — always closer, never equal. Arctangent is odd, so arctan(−1) = −45°. The Wolfram MathWorld entry on the inverse tangent covers the function and its series expansion.
The Quadrant Problem, and Why atan2 Exists
This is the practical failure mode of plain arctangent, and it is worth understanding properly because it produces bugs that are hard to see. Tangent repeats every 180°, not 360°, because it is a ratio of two coordinates and flipping the sign of both leaves the ratio unchanged. So the point (4, 3) and the point (−4, −3) — which point in exactly opposite directions — have the identical tangent of 0.75.
Once you have divided y by x, that information is gone. Arctangent sees 0.75 and returns 36.87°, and it has no way to know whether you meant 36.87° or 216.87°. It always chooses the first- or fourth-quadrant answer. In a navigation, robotics or graphics context this means a heading that is precisely 180° wrong, which typically shows up as an object moving backwards or a robot turning the wrong way — a bug that looks bizarre until you trace it to a division.
The fix is atan2, which takes y and x as two separate arguments and never performs the destructive division. Because it can inspect both signs, it distinguishes all four quadrants and returns an angle across the full turn, conventionally from −180° to 180°. Working the loaded example: y = −3, x = −4 puts the point in the third quadrant, the reference angle is arctan(3⁄4) = 36.87°, and atan2 returns −143.13°, equivalently 216.87° measured anticlockwise from the positive x‑axis. It also handles x = 0 gracefully, returning 90° or −90° where a plain division would fail. The reference angle calculator shows the acute-angle half of this relationship directly.
Slopes, Gradients and Percentage Grades
Arctangent is the tool that converts a slope into an angle. A road with a 10% grade rises 10 units for every 100 travelled horizontally, so the angle is arctan(0.10) = 5.71°. A 100% grade is not vertical, as people often assume — it is arctan(1) = 45°. The relationship is strongly non-linear at the top: a 200% grade is 63.4°, and no finite percentage ever reaches 90°, which is the asymptote showing up in an everyday setting.
The same conversion links coordinate geometry to angle geometry. A line of slope m makes an angle arctan(m) with the horizontal, which is how our slope calculator connects to this page: it gives you m from two points, and arctangent turns m into an inclination. Roof pitches quoted as "6 in 12" are arctan(6⁄12) = 26.57°. Camera field of view, wheelchair ramp compliance angles and screen viewing angles are all arctangent calculations on a ratio of two lengths.
Exact Values and the Asymptote
The exact values are fewer than for sine and cosine but just as worth knowing: arctan(0) = 0, arctan(√3⁄3) = π⁄6 (30°), arctan(1) = π⁄4 (45°), arctan(√3) = π⁄3 (60°). Negatives mirror them. Note that √3⁄3 is 0.5773502692 — the same number as 1⁄√3, written with a rationalised denominator.
The asymptote has one genuinely useful property: as x grows large, arctan(x) approaches 90° so gently that huge changes in input produce tiny changes in output. Going from x = 1,000 to x = 10,000 moves the angle by less than a hundredth of a degree. This makes arctangent a natural "squashing" function — it maps the entire infinite real line into a bounded interval smoothly and reversibly, which is why it appears as an activation function in machine learning and as a soft limiter in signal processing. It also means that recovering a large tangent value back from an angle near 90° is numerically fragile: a rounding error of 0.01° near the asymptote can change the tangent by hundreds. The trigonometric functions calculator shows how quickly tangent climbs as you approach 90°.
Why Arctangent Never Rejects an Input
Sine and cosine are coordinates on a circle of radius 1, so they are permanently bounded by ±1 and their inverses reject anything outside that band. Tangent is not a coordinate; it is a ratio of two coordinates, and a ratio has no ceiling once the denominator is free to approach zero. Geometrically, tangent measures the length cut off on a line drawn tangent to the circle, and that line runs to infinity in both directions.
The consequence for everyday use is that an arctangent calculation never fails on domain grounds, which makes it the more robust choice when a ratio might be extreme. Where arcsine would reject 1.4 outright, arctangent takes it happily and returns 54.46°. The trade-off is that the answer is only ever the principal value, so you carry the quadrant responsibility yourself unless you use atan2. The NIST Digital Library of Mathematical Functions documents the domains and branch cuts of all the inverse trigonometric functions, and the angle converter handles any unit conversion the result needs.
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Browse All Free Tools Contact Arb DigitalCommon Mistakes to Avoid
- Dividing y by x before taking the arctangent — that division destroys the quadrant. Use atan2 with the two components whenever direction matters.
- Expecting arctan to reach 90° — it never does. The range is open, and any answer of exactly 90° means something rounded.
- Assuming a 100% grade is vertical — it is 45°. Grade percentages and angles are not proportional.
- Reversing atan2's arguments — it is atan2(y, x), with the vertical component first. The reversed order returns the complementary angle.
- Trusting a tangent near the asymptote — a small angular rounding error becomes an enormous change in the tangent value above roughly 85°.
Related Free Tools From Arb Digital
Compare ranges with the arcsin calculator and the arccos calculator, get a slope from two points with the slope calculator, evaluate tangent forwards on the trigonometric functions calculator, strip an angle to its acute equivalent with the reference angle calculator, or browse the whole free online tools hub.
Frequently Asked Questions
(−π⁄2, π⁄2), or −90° to 90° exclusive. The endpoints are approached as the input grows without bound but are never actually reached, which is why the interval is open.
Sine and cosine are coordinates on a circle of radius 1, so they are bounded by ±1. Tangent is the ratio of those two coordinates, and a ratio grows without limit as its denominator approaches zero, so every real number is the tangent of some angle.
atan2(y, x) takes the two components separately instead of their ratio, so it can read both signs and return the correct angle in any of the four quadrants across a full 360°. Use it whenever direction matters, such as bearings, headings or polar coordinates.
Because you divided y by x before taking the arctangent. That division loses the sign information, so (4, 3) and (−4, −3) produce the identical ratio despite pointing in opposite directions. Use atan2 instead.
Divide the percentage by 100 and take the arctangent. A 10% grade is arctan(0.10) = 5.71°, and a 100% grade is arctan(1) = 45°, not vertical.
π⁄4 radians, which is 45° — the angle where rise and run are equal. arctan(√3) is π⁄3 (60°) and arctan(√3⁄3) is π⁄6 (30°).
No. arctan is the inverse function of tangent. The reciprocal of tangent is cotangent, an entirely different function. The superscript −1 in tan-1 means inverse, not reciprocal.
This tool is provided for education and reference. Verify any result used in navigation, engineering or construction work against an independent calculation.
